Motivated by recent observations which detect an outer boundary for starless cores , and evidence for time-dependent mass accretion in the Class 0 and Class I protostellar phases , we reexamine the case of spherical isothermal collapse in the case of a finite mass reservoir .
The presence of a core boundary results in the generation of an inward propagating rarefaction wave .
This steepens the gas density profile from r ^ { -2 } to r ^ { -3 } or steeper .
After a protostar forms , the mass accretion rate \dot { M } evolves through three distinct phases : ( 1 ) an early phase of decline in \dot { M } , which is a non-self-similar effect due to spatially nonuniform infall in the prestellar phase ; ( 2 ) for large cores , an intermediate phase of near-constant \dot { M } from the infall of the outer part of the self- similar density profile ; ( 3 ) a late phase of rapid decline in \dot { M } when accretion occurs from the region affected by the inward propagating rarefaction wave .
Our model clouds of small to intermediate size make a direct transition from phase ( 1 ) to phase ( 3 ) above .
Both the first and second phase are characterized by a temporally increasing bolometric luminosity L _ { bol } , while L _ { bol } is decreasing in the third ( final ) phase .
We identify the period of temporally increasing L _ { bol } with the Class 0 phase , and the later period of terminal accretion and decreasing L _ { bol } with the Class I phase .
The peak in L _ { bol } corresponds to the evolutionary time when 50 \% \pm 10 \% of the cloud mass has been accreted by the protostar .
This is in agreement with the classification scheme proposed by André et al .
( 1993 ) .
We show how our results can be used to explain tracks of envelope mass M _ { env } versus L _ { bol } for protostars in Taurus and Ophiuchus .
We also develop an analytic formalism which reproduces the protostellar accretion rate .